Problem using FFT to get prob dist in Matlab

[jellysky]

I am trying to use the Fast Fourier Transform to derive the probability distribution for an affine jump diffusion model. For some reason, I am having an enormously difficult time doing this in Matlab. I have posted my code below for trying to derive the Gaussian distribution from the Gaussian characteristic function (sort of like the Moment Generating Function). However I need to divide the resultant pdf by the interval size and still I get too much probability in the extremes (I keep getting probability's of 0.025 out for x's > 5 and x's < -5). Can someone please help me figure out what I am doing wrong.....Thanks.

In addition I have attached an excel file that contains the FFT resultant pdf and the correct pdf as generated by Matlab's 'pdf' function.

l = -10;
u = 10;
T = 512;
dx = (u-l)/T;
n = [0:T-1];
x = (l)+n*dx; %my pdf x's
t = [0:1:T-1];
s_t = 2*pi*t/(T*dx); %my s's for the Gaussian characteristic function
mu = 0;
sigma = 1;
lambda = 2;

norm = exp(s_t.*i.*mu - 0.5.*(sigma.*s_t).^2 - i.*s_t.*l); %Gaussian characteristic function (MGF except with 'it' substituted for 't'
pdfnorm = real(fft(norm))./(u-l);
plot(x,pdfnorm); %the whole distribution is wrong....it has too much probability at the extremes...and it does this even when I increase T or increase l and u...HELP!!!


This is what the pdf and psi function should look like

xlr=10; np=512; dx=2*xlr/np;
x=dx*[0:np-1]-xlr; % OR, equivalently
x=linspace(-xlr,xlr-dx,np);

normp1 = pdf('norm',x,0,1);
fnormp1 = fft(normp1);
normp1after = ifft(fnormp1);
plot (x,normp1)

 

[Shlomi]

Its been a while since Ive done this things but I believe that your problem is that you missing fftshift after the fft.
After you do fft the zero-frequency component isnt in the middle of the spectrum and you need to move it.
Im not sure if thats your problem, but from my signal processing background I can tell that this looks fishy.
After fft you have to do fftshift and after ifft you have to do ifftshift.
Try that.